\(\int \frac {\arcsin (a x)^4}{x} \, dx\) [35]

Optimal result

Integrand size = 10, antiderivative size = 113 \[ \int \frac {\arcsin (a x)^4}{x} \, dx=-\frac {1}{5} i \arcsin (a x)^5+\arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+3 \arcsin (a x)^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )+3 i \arcsin (a x) \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (5,e^{2 i \arcsin (a x)}\right ) \] Output:

-1/5*I*arcsin(a*x)^5+arcsin(a*x)^4*ln(1-(I*a*x+(-a^2*x^2+1)^(1/2))^2)-2*I* 
arcsin(a*x)^3*polylog(2,(I*a*x+(-a^2*x^2+1)^(1/2))^2)+3*arcsin(a*x)^2*poly 
log(3,(I*a*x+(-a^2*x^2+1)^(1/2))^2)+3*I*arcsin(a*x)*polylog(4,(I*a*x+(-a^2 
*x^2+1)^(1/2))^2)-3/2*polylog(5,(I*a*x+(-a^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)^4}{x} \, dx=\frac {1}{5} i \arcsin (a x)^5+\arcsin (a x)^4 \log \left (1-e^{-2 i \arcsin (a x)}\right )+2 i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (a x)}\right )+3 \arcsin (a x)^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (a x)}\right )-3 i \arcsin (a x) \operatorname {PolyLog}\left (4,e^{-2 i \arcsin (a x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (5,e^{-2 i \arcsin (a x)}\right ) \] Input:

Integrate[ArcSin[a*x]^4/x,x]
 

Output:

(I/5)*ArcSin[a*x]^5 + ArcSin[a*x]^4*Log[1 - E^((-2*I)*ArcSin[a*x])] + (2*I 
)*ArcSin[a*x]^3*PolyLog[2, E^((-2*I)*ArcSin[a*x])] + 3*ArcSin[a*x]^2*PolyL 
og[3, E^((-2*I)*ArcSin[a*x])] - (3*I)*ArcSin[a*x]*PolyLog[4, E^((-2*I)*Arc 
Sin[a*x])] - (3*PolyLog[5, E^((-2*I)*ArcSin[a*x])])/2
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.31, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5136, 3042, 25, 4200, 25, 2620, 3011, 7163, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arcsin (a x)^4}{x} \, dx\)

\(\Big \downarrow \) 5136

\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \arcsin (a x)^4}{a x}d\arcsin (a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\arcsin (a x)^4 \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \arcsin (a x)^4 \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)\)

\(\Big \downarrow \) 4200

\(\displaystyle 2 i \int -\frac {e^{2 i \arcsin (a x)} \arcsin (a x)^4}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 25

\(\displaystyle -2 i \int \frac {e^{2 i \arcsin (a x)} \arcsin (a x)^4}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 2620

\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \int \arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 3011

\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \left (\frac {1}{2} i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \int \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )\right )-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 7163

\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \left (\frac {1}{2} i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \left (i \int \arcsin (a x) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )d\arcsin (a x)-\frac {1}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )\right )\right )\right )-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 7163

\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \left (\frac {1}{2} i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \left (i \left (\frac {1}{2} i \int \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )d\arcsin (a x)-\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )\right )-\frac {1}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )\right )\right )\right )-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 2720

\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \left (\frac {1}{2} i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \left (i \left (\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}-\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )\right )-\frac {1}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )\right )\right )\right )-\frac {1}{5} i \arcsin (a x)^5\)

\(\Big \downarrow \) 7143

\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^4 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 i \left (\frac {1}{2} i \arcsin (a x)^3 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \left (i \left (\frac {1}{4} \operatorname {PolyLog}\left (5,e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )\right )-\frac {1}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )\right )\right )\right )-\frac {1}{5} i \arcsin (a x)^5\)

Input:

Int[ArcSin[a*x]^4/x,x]
 

Output:

(-1/5*I)*ArcSin[a*x]^5 - (2*I)*((I/2)*ArcSin[a*x]^4*Log[1 - E^((2*I)*ArcSi 
n[a*x])] - (2*I)*((I/2)*ArcSin[a*x]^3*PolyLog[2, E^((2*I)*ArcSin[a*x])] - 
((3*I)/2)*((-1/2*I)*ArcSin[a*x]^2*PolyLog[3, E^((2*I)*ArcSin[a*x])] + I*(( 
-1/2*I)*ArcSin[a*x]*PolyLog[4, E^((2*I)*ArcSin[a*x])] + PolyLog[5, E^((2*I 
)*ArcSin[a*x])]/4))))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( 
a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.54

method	result	size
derivativedivides	\(-\frac {i \arcsin \left (a x \right )^{5}}{5}+\arcsin \left (a x \right )^{4} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 i \arcsin \left (a x \right )^{3} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+12 \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 i \arcsin \left (a x \right ) \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{4} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \arcsin \left (a x \right )^{3} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+12 \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+24 i \arcsin \left (a x \right ) \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, i a x +\sqrt {-a^{2} x^{2}+1}\right )\)	\(287\)
default	\(-\frac {i \arcsin \left (a x \right )^{5}}{5}+\arcsin \left (a x \right )^{4} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 i \arcsin \left (a x \right )^{3} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+12 \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 i \arcsin \left (a x \right ) \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{4} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \arcsin \left (a x \right )^{3} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+12 \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+24 i \arcsin \left (a x \right ) \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, i a x +\sqrt {-a^{2} x^{2}+1}\right )\)	\(287\)

Input:

int(arcsin(a*x)^4/x,x,method=_RETURNVERBOSE)
 

Output:

-1/5*I*arcsin(a*x)^5+arcsin(a*x)^4*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))-4*I*arcs 
in(a*x)^3*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+12*arcsin(a*x)^2*polylog(3, 
-I*a*x-(-a^2*x^2+1)^(1/2))+24*I*arcsin(a*x)*polylog(4,-I*a*x-(-a^2*x^2+1)^ 
(1/2))-24*polylog(5,-I*a*x-(-a^2*x^2+1)^(1/2))+arcsin(a*x)^4*ln(1-I*a*x-(- 
a^2*x^2+1)^(1/2))-4*I*arcsin(a*x)^3*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+12 
*arcsin(a*x)^2*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))+24*I*arcsin(a*x)*polylo 
g(4,I*a*x+(-a^2*x^2+1)^(1/2))-24*polylog(5,I*a*x+(-a^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \frac {\arcsin (a x)^4}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x} \,d x } \] Input:

integrate(arcsin(a*x)^4/x,x, algorithm="fricas")
 

Output:

integral(arcsin(a*x)^4/x, x)
 

Sympy [F]

\[ \int \frac {\arcsin (a x)^4}{x} \, dx=\int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x}\, dx \] Input:

integrate(asin(a*x)**4/x,x)
 

Output:

Integral(asin(a*x)**4/x, x)
 

Maxima [F]

\[ \int \frac {\arcsin (a x)^4}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x} \,d x } \] Input:

integrate(arcsin(a*x)^4/x,x, algorithm="maxima")
 

Output:

integrate(arcsin(a*x)^4/x, x)
 

Giac [F]

\[ \int \frac {\arcsin (a x)^4}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x} \,d x } \] Input:

integrate(arcsin(a*x)^4/x,x, algorithm="giac")
 

Output:

integrate(arcsin(a*x)^4/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^4}{x} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x} \,d x \] Input:

int(asin(a*x)^4/x,x)
 

Output:

int(asin(a*x)^4/x, x)
 

Reduce [F]

\[ \int \frac {\arcsin (a x)^4}{x} \, dx=\int \frac {\mathit {asin} \left (a x \right )^{4}}{x}d x \] Input:

int(asin(a*x)^4/x,x)
 

Output:

int(asin(a*x)**4/x,x)